TITLE: Yoneda-Lemma as generalization of Cayley`s theorem?
QUESTION [49 upvotes]: I came across the statement that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations.
How exactly is Yoneda-lemma a generalization of Cayley`s theorem? Can Cayley's theorem be deduced from Yoneda lemma, is it a generalization of a particular case of Yoneda, or is this instead, a philosophical statement?
To me, it seems that Yoneda embedding is more canonical than Cayley's theorem because in the latter you have to choose whether the group acts from the left or from the right on itself. But maybe this is an optical illusion.

REPLY [22 votes]: I can also add that both Yoneda and Cayley are results which follow from the general philosophy of investigating algebraic structures by letting them act on themselves.
1) If you let a group $G$ act on itself, you realize it as a subgroup of the permutation group of the underlying set; in particular if $G$ is finite, as a subgroup of $\mathfrak{S}_n$.
2) If you let a ring with unit act on itself, you realize it as a subring of $\operatorname{End}(E)$, where $E$ is the underlying additive group.
3) Similarly, if you let a finite dimensional $k$-algebra act on itself, you realize it as a subalgebra of the matrix algebra $\mathcal{M}_n(k)$. In particular this gives the classical realization of $\mathbb{C}$ as a matrix algebra over $\mathbb{R}$ and of the quaternions as a matrix algebra over $\mathbb{C}$ or $\mathbb{R}$.
4) You can let a Lie algebra act on itself, but unfortunately this action need not be faithful (Lie algebra don't have units...). So you only obtain the easy first step of Ado's theorem about embedding Lie algebras into matrix Lie algebras.
5) If you let a category $\mathcal{C}$ act on itself, you obtain an embedding into $Fun(\mathcal{C}^{op}, Set)$, which is the content of Yoneda's lemma.